Page:TolmanEquations.djvu/7

 Gravitational Field.

This method of obtaining from Coulomb’s law the expected expression for the force exerted by a moving electric charge is of special interest, since it suggests the possibility of obtaining from Newton’s law an expression for the gravitational force exerted by a moving mass.

Let us assume, in accordance with Newton’s law, that a stationary mass $$m_1$$ will act on any other mass $$m_2$$ with the force $$F=-km_{1}m_{2}\frac{\mathsf{r}}{r^{3}}$$, where $$m_1$$ and $$m_2$$ are the masses which the particles would have if they were at rest, isolated, and at the absolute zero of temperature, and r the radius vector from $$m_1$$ to $$m_2$$. The determination of the force exerted by a mass in uniform motion may now be carried out in exactly the same manner as for the force exerted by a moving charge. In fact in analogy to equations (24), (25), and (26), we may write–

These are the components of the force with which a particle of "stationary" mass $$m_1$$, in uniform motion in the X direction with the velocity $$v$$, acts on another particle of "stationary" mass $$m_2$$. Taking $$m_1$$ as the centre of coordinates, $$m_2$$ has the coordinates X, Y, and Z and the velocity $$\left(\mathsf{\dot{X},\dot{Y},\dot{Z}}\right)$$. $$k$$ is the constant of gravitation, $$\beta$$ is placed equal to $$\tfrac{v}{c}$$, and $$s$$ has been substituted for $$\sqrt{\mathsf{X}^{2}+\left(1-\beta^{2}\right)\left(\mathsf{Y}^{2}+\mathsf{Z}^{2}\right)}$$.

It may be noted that the particle $$m_1$$ must be in uniform motion, although the particle $$m_2$$ may have any motion, its instantaneous velocity being $$\left(\mathsf{\dot{X},\dot{Y},\dot{Z}}\right)$$. It is unfortunate that the method does not also permit a determination of the force which an accelerated particle exerts. For cases, however, where the acceleration is slow enough to be neglected, it would